the setpoint overshoot method: a simple and fast closed-loop...

The setpoint overshoot method: A simple and fast closed-loop approach for PID tuning*

Mohammad Shamsuzzoha, Sigurd Skogestad†

Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway

Abstract:

A simple method has been developed for PID controller tuning of an unidentified

process using closed-loop experiments. The proposed method requires one closed-loop

step setpoint response experiment using a proportional only controller, and it mainly

uses information about the first peak (overshoot) which is very easy to identify. The

setpoint experiment is similar to the classical Ziegler-Nichols (1942) experiment, but

the controller gain is typically about one half, so the system is not at the stability limit

with sustained oscillations. Based on simulations for a range of first-order with delay

processes, simple correlations have been derived to give PI controller settings similar to

those of the SIMC tuning rules (Skogestad, 2003). The recommended controller gain

change is a function of the height of the first peak (overshoot), whereas the controller

integral time is mainly a function of the time to reach the peak. The method includes a

detuning factor that allows the user to adjust the final closed-loop response time and

robustness. The proposed tuning method, originally derived for first-order with delay

processes, has been tested on a wide range of other processes typical for process control

applications and the results are comparable with the SIMC tunings using the open-loop

model.

* This is an extended version of a paper presented at the IFAC Symposium on Dynamics and Control of Process Systems (DYCOPS) in Belgium in July 2010. † Correspondence author. Email: [email protected]

1 Introduction

The proportional integral (PI) controller is widely used in the process industries due to

its simplicity, robustness and wide ranges of applicability in the regulatory control

layer. On the basis of a survey of more than 11 000 controllers in the process industries,

Desborough and Miller [1] report that more than 97% of the regulatory controllers

utilise the PID algorithm. A recent survey (Kano and Ogawa [2]) from Japan shows that

the ratio of applications of PID control, conventional advanced control (feedforward,

override, valve position control, gain-scheduled PID, etc.) and model predictive control

is about 100:10:1. In addition, the vast majority of the PID controllers do not use

derivative action. Even though the PI controller only has two adjustable parameters, it is

not simple to find good settings and many controllers are poorly tuned. One reason is

that quite tedious plant tests may be needed to obtain improved controller settings. The

objective of this paper is to derive a method which is simpler to use than the present

ones.

Most tuning approaches are based on an open-loop plant model (g); typically given in

terms of the plant’s gain (k), time constant (τ) and time delay (θ); see O’Dwyer [3] for

an extensive list of methods. Given a plant model g, one popular approach to obtain the

controller is direct synthesis (Seborg et al., [4]) which includes the IMC-PID tuning

method of Rivera et al. [5]. The original direct synthesis approaches, like that of Rivera

et al. [5], give very good performance for setpoint changes, but give sluggish responses

to input (load) disturbances for lag-dominant (including integrating) processes with τ/θ

larger than about 10. To improve load disturbance rejection, Skogestad [6] proposed the

modified SIMC method where the integral time is reduced for processes with a large

value of the process time constant τ. The SIMC rule has one tuning parameter, the

closed-loop time constant τc, and for “fast and robust” control is recommended to

choose τc= θ, where θ is the (effective) time delay.

However, these approaches require that one first obtains an open-loop model (g) of the

process. There are two problems here. First, an open-loop experiment, for example a

step test, is normally needed to get the required process data. This may be time

consuming and may result in undesirable output changes. Second, approximations are

involved in obtaining the process model g from the open-loop data.

In this paper, the objective is to derive controller tunings based on closed-loop

experiments. The simplest is to directly obtain the controller from the closed-loop data,

without explicitly obtaining an open-loop model g. This is the approach of the classical

Ziegler-Nichols method [7] which requires very little information about the process;

namely, the ultimate controller gain (Ku) and the period of oscillations (Pu) which are

obtained from a single experiment. For a PI-controller the recommended settings are

Kc=0.45Ku and τI=0.83Pu. However, there are several disadvantages. First, the system

needs to be brought its limit of instability and a number of trials may be needed to bring

the system to this point. To avoid this problem one may induce sustained oscillation

with an on-off controller using the relay method of Åström and Hägglund, [8].

However, this requires that the feature of switching to on/off-control has been installed

in the system. Another disadvantage is that the Ziegler-Nichols [7] tunings do not work

well on all processes. It is well known that the recommended settings are quite

aggressive for lag-dominant (integrating) processes (Tyreus and Luyben, [9]) and quite

slow for delay-dominant process (Skogestad, [6]). To get better robustness for the lag-

dominant (integrating) processes, Tyreus and Luyben [9] proposed to use less

aggressive settings (Kc=0.313Ku and τI=2.2Pu), but this makes the response even slower

for delay-dominant processes (Skogestad, [6]). This is a fundamental problem of the

Ziegler-Nichols [7] method because it uses only two pieces of information about the

process (Ku, Pu), which correspond to the critical point on the Nyquist curve. This does

allow one to distinguish, for example, between a lag-dominant and a delay-dominant

process. A fix is to use additional closed-loop experiments, for example an experiment

with an integrating controller (Schei [15]). A third disadvantage of the Ziegler-Nichols

[7] method is that it can only be used on processes for which the phase lag exceeds -180

degrees at high frequencies. For example, it does not work on a simple second-order

process.

Therefore, there is need of an alternative closed-loop approach for plant testing and

controller tuning which avoids the instability concern during the closed-loop

experiment, reduces the number of trails, and works for a wider range of processes. The

proposed new method satisfies these concerns:

1. The method uses a single closed-loop experiment with proportional only control.

This is similar to the Ziegler-Nichols [7] method, but the process is not forced to its

stability limit and it requires less trial-and-error adjustment of the P-controller gain

to get to the desired closed-loop response.

2. Of the many parameters that can be obtained from the closed-loop setpoint response,

the simplest to observe is the time (tp) and magnitude (overshoot) of the first peak

(see Figure 1) which is the main information used in the proposed method.

3. The proposed method works well on a wider range of processes than the Ziegler-

Nichols [7] method. In particular, it works well also for delay-dominant processes.

This is because it that makes use of a third piece of information, namely the relative

steady-state change b = y(∞)/ys.

4. The method applies to processes that give overshoot with proportional only control.

This is less restrictive than the Ziegler-Nichols [7] method, which requires sustained

oscillations. Thus, unlike the Ziegler-Nichols method, the method works on a simple

second-order process.

In summary, the proposed method is simpler in use than existing approaches and allows

the process to be kept under closed-loop control.

2. SIMC PI tuning rules

In Figure 2 we show the block diagram of a conventional feedback control system,

where g denotes the process transfer function and c the feedback controller. The other

variables are the manipulated variable u, the measured and controlled output variable y,

the setpoint ys, and the disturbance d which is here assumed to be a “load disturbance”

at the plant input. The closed-loop transfer functions from the setpoint and load

disturbance to the output are:

s

cg gy= y + d

1+cg 1+cg (1)

In process control, a first-order process with time delay is a common representation of

the process dynamics:

-θskeg(s)=

τs+1 (2)

Here k is the process gain, τ the dominant lag time constant and θ the effective time

delay. Most processes in the process industries can be satisfactorily controlled using a

PI controller:

cI

1c s =K 1+

τ s

(3)

which in the time domain corresponds to

t

cc

I 0

Ku t =K e t + e t dt

τ (4)

where e = ys-y. The PI controller has two adjustable parameters, the proportional gain

Kc and the integral time τI. The ratio I c IK =K τ is known as the integral gain.

The SIMC tuning rule (Skogestad, [6]) is analytically based and widely used in the

process industry. For the process in Eq. (2), the SIMC tuning rule gives

cc

τK =

k τ +θ (5)

I cτ =min τ, 4(τ +θ) (6)

Note that the original IMC tuning rule (Rivera et al., [5]) always uses τI = τ, but the

SIMC rule increases the integral contribution for close-to integrating processes (with τ

large) to avoid poor performance (slow settling) to load disturbance. There is one

adjustable tuning parameter, the closed-loop time constant (τc), which is selected to give

the desired trade-off between performance and robustness. Initially, this study is based

on the “fast and robust” setting

cτ =θ (7)

which gives a good trade-off between performance and robustness. In terms of

robustness, this choice gives a gain margin is about 3 and a sensitivity peak (Ms-value)

of about 1.6. On dimensionless form, the SIMC tuning rules with τc = θ become

'c c

τK =kK =0.5

θ (8)

' cI

I

kK 1 τK = =max 0.5,

τ θ 16 θ

(9)

The dimensionless gains 'cK and '

IK are plotted as a function of τ/θ in Figure 3. We

note that the integral term ( 'IK ) is relatively more important for delay dominant

processes (τ/θ<1), while the proportional term 'cK is more significant for processes with

a smaller time delay. These insights are useful for the next step when we want to derive

tuning rules based on the closed-loop setpoint response.

3. Closed-loop setpoint experiment

As mentioned earlier, the objective is to base the controller tuning on closed-loop data.

The simplest closed-loop experiment is probably a setpoint step response where one

maintains control of the process, including the change in the output variable. From the

setpoint experiment (Figure 1) one may observe many values, like rise time, period of

oscillations, magnitudes and times of overshoots and undershoots, etc. Of all these

values, the simplest to observe is the magnitude and time (tp) of the (first) overshoot,

and this information is therefore the basis for the proposed method.

We propose the following procedure:

Step 1. Switch the controller to P-only mode (for example, increase the integral time τI

to its maximum value or set the integral gain KI to zero). In an industrial system, with

bumpless transfer, the switch should not upset the process.

Step 2. Make a setpoint change with a P-only controller. The P-controller gain Kc0 used

in the experiment does not really matter as long as the response oscillates sufficiently

with an overshoot between 0.10 (10%) and 0.60 (60%); about 0.30 (30%) is a good

value. Most likely, unless the original controller was tightly tuned, one will need to

increase the controller gain to get a sufficiently large overshoot. Note that the controller

gain to get 30% overshoot is about half of the “ultimate” controller gain needed in the

Ziegler-Nichols closed-loop experiment.

Step 3. From the closed-loop setpoint response experiment, obtain the following values

(see Figure 3):

Controller gain, Kc0

Overshoot = (Δyp - Δy∞) /Δy∞

Time from setpoint change to reach peak output (overshoot), tp

Relative steady state output change, b = Δy∞/Δys.

Here the output variable changes are:

sΔy = ys – y0: Setpoint change

pΔy = yp – y0: Peak output change (at time tp)

Δy = y∞ - y0: Steady-state output change after setpoint step test

To find Δy∞ one needs to wait for the response to settle, which may take some time if

the overshoot is relatively large (typically, 0.3 or larger). In such cases, one may stop

the experiment when the setpoint response reaches its first minimum (undershoot) and

record the corresponding output, uΔy . As shown in Appendix A, one can then estimate

the steady-state change from the following correlation:

Δy∞ = 0.45(Δyp + Δyu) (10)

Note that (10) involves deviations from the original steady state y0; in terms of the

actual variables we have y∞ = 0.45(yp + yu) + 0.1 y0.

To illustrate the use of the closed-loop setpoint experiment, we show in Figure 4 closed-

loop responses for a typical process with a unit time delay (θ=1) and a ten times larger

time constant (τ=10): -se

g(s)=10s+1

(11)

The responses in Figure 4 are for six different controller gains Kc0, which result in

overshoots of 0.10, 0.20, 0.30, 0.40, 0.50 and 0.60, respectively. As expected, the

closed-loop response gets faster and more oscillatory as the overshoot increases. Note

that small overshoots (less than 0.10) are not shown. The main reason is that it is

difficult in practice to obtain from experimental data accurate values of the overshoot

and corresponding time if the overshoot is too small. Also, large overshoots (larger than

about 0.6) are not shown, because these give a long settling time and require more

excessive input changes. For these reasons we recommend using an “intermediate”

overshoot of about 0.3 (30%) for the closed-loop setpoint experiment.

Figure 5 shows setpoint responses when the P-controller gain Kc0 has been adjusted to

give an overshoot of 0.3 for a wide range of first-order plus delay processes with a unit

time delay (θ=1), -se

g(s)=τs+1

(12)

The process time constant τ varies from 0 (pure delay process) to 100 (almost

integrating process). The time to reach the first peak (tp) increases somewhat as we

increase τ, but the most striking difference is that the steady-state output change (b-

value) approaches 1 as we increase τ. Thus, the b-value provides an indirect measure of

the value of τ/θ, and we will make use of this observation below.

4. Correlation between setpoint response and SIMC-settings

As mentioned in the introduction, a two-step procedure could be used where first the

closed-loop setpoint response experiment is used to determine the open-loop model

parameters (k, τ, θ) and next the SIMC-rules (or others) are used to derive PI-settings.

However, the objective of this paper is to provide a more direct approach similar to the

Ziegler-Nichols [7] closed-loop method.

Thus, the goal is to derive a correlation, preferably as simple as possible, between the

setpoint response data (Figure 1) and the SIMC PI-settings in Eq. (5) and (6), initially

with the choice τc=θ. For this purpose, we considered 15 first-order with delay models -θske

g(s)=τs+1

(13)

that cover a wide range of processes; from delay-dominant to lag-dominant

(integrating):

τ θ =0.1, 0.2, 0.4, 0.8, 1.0, 1.5, 2.0, 2.5,3.0, 5.0, 7.5, 10.0, 20.0, 50.0, 100.0

Since we can always scale time with respect to the time delay (θ) and since the closed-

loop response depends on the product of the process and controller gains (kKc) we have

without loss of generality used in all simulations k=1 and θ=1.

For each of the 15 process models (values of τ/θ), we obtained the SIMC PI-settings (Kc

and τI) using Eqs. (5) and (6) with the choice τc=θ. Furthermore, for each of the 15

processes we generated 6 closed-loop step setpoint responses (Figure 4 and Figure 5)

using P-controllers that give different fractional overshoots.

Overshoot= 0.10, 0.20, 0.30, 0.40, 0.50 and 0.60

In total, we then have 90 setpoint responses, and for each of these we record four data:

the P-controller gain Kc0 used in the experiment, the fractional overshoot, the time to

reach the overshoot (tp), and the relative steady-state change, b = Δy∞/Δys.

Controller gain (Kc). We first seek a relationship between the above four data and the

corresponding SIMC-controller gain Kc. Recall that with PI-control, the recommended

Ziegler-Nichols [7] controller gain for any process is Kc/Ku = 0.45, where Ku is the

“ultimate” controller gain that gives persistent oscillations in the Ziegler-Nichols

experiment. As mentioned, this is generally viewed to be too aggressive and Tyreus and

Luyben [9] recommend Kc/Ku = 0.31. Note that Ku is similar to our Kc0, and since the

Ziegler-Nichols experiment is similar to a setpoint response with about 100%

overshoot, one may hope that we may use a similar simple relationship. Indeed, as

illustrated in Figure 6, where we plot kKc (scaled SIMC PI-controller gain for the

process) as a function of kKc0 (scaled experimental controller gain for the same process)

for the 90 setpoint experiments, the ratio

c

c0

K=A

K (14)

is approximately constant for a fixed value of the overshoot, independent of the value of

τ/θ. In Figure 7 we plot the value of A, obtained as the best fit of the slopes of the lines

in Figure 6, as a function of the overshoot. The ratio A is found to vary from 0.85 for

overshoot=0.1, to 0.62 for overshoot=0.3, and 0.45 for overshoot=0.6. The following

equation (solid line in Figure 7) fits the data well,

2A= 1.152(overshoot) - 1.607(overshoot) + 1.0 (15)

The correlation in Eq.(15) is based on data with overshoots between 0.1 and 0.6 and

should not be extended outside this range. To compare with the Ziegler-Nichols [7]

ultimate gain approach, note that a value of A of about 0.31 (Tyreus and Luyben [9])

seems reasonable if we imagine extending Figure 7 to overshoots over 100%.

Actually, a closer look at Figure 6 reveals that a constant slope, use of Eq.(14) and (15),

only fits the data well for a scaled controller gain 'c cK =kK greater than about 0.5.

Fortunately, a good fit of the controller gain Kc is not so important for delay-dominant

processes (τ/θ<1) where 'cK <0.5 , because we recall from the discussion of the SIMC

rules (Figure 2) that the integral gain KI is more important for such processes. This is

discussed in more detail below.

Integral time (τI). Next, we want to find a simple correlation for the integral time. We

follow the SIMC tuning formula in Eq. (6) and use the minimum from two cases:

Case (1). τI1 =τ for processes with a relatively large delay θ.

Case (2). τI2 =4(τc+θ) for processes with a relatively small delay θ including

integrating processes. In the following we consider the nominal choice τc=θ

which gives τI2 =8θ, but we will later show how to reintroduce a detuning factor.

Case (1). We do not know the value of the dominant time constant τ. However, τ

enters into the SIMC rule for the controller gain in Eq. (8), so we can insert τ = τI1 into

Eq. (8) and solve for τI1 to get:

I1 cτ =2kK θ (16)

Rewrite kKc as

kKc = Kc/Kc0 · k Kc0 (17)

Here, Kc /Kc0 =A where A is given as a function of the overshoot in Eq. (15). The value

of the loop gain kKc0 for the P-control setpoint experiment is given from the value of b:

c0

bkK =

(1-b) (18)

To prove this, note from Eq.(1) that the closed-loop setpoint response is Δy/Δys =

gc/(1+gc) and with a P-controller with gain Kc0, the steady-state value is Δy∞/Δys =

kKc0/(1+kKc0)=b and we derive Eq.(18). The absolute value is included to avoid

problems if b>1, as may occur because of inaccurate data or for an unstable process.

To sum up, we have derived the following expression for the integral time for a delay-

dominant process (with τ/θ<8):

I1

bτ =2A θ

1-b (19)

Case (2). For a lag-dominant (including integrating) process with τ/θ>8, the nominal

SIMC integral time SIMC is

τI2=8θ (20)

Equations (19) and (20) for the integral time have all known parameters except the

effective time delay θ. One could obtain θ the directly from the closed-loop setpoint

experiment, but this is generally not easy. Fortunately, as shown in Figure 8, there is a

reasonably good correlation between θ and the setpoint peak time tp which is much

easier to observe.

Case (1). For processes with a relatively large time delay (τ/θ<8), the ratio θ/tp varies

between 0.27 (for τ/θ= 8 with overshoot=0.1) and 0.5 (for τ/θ=0.1 with all overshoots).

For the intermediate overshoot of 0.3, the ratio θ/tp varies between 0.32 and 0.50. A

conservative choice would be to use θ=0.5tp because this gives the largest integral time.

However, to improve performance for processes with smaller time delays, we propose

to use θ=0.43tp which is only 14% lower than 0.50 (the worst case).

In conclusion, we have for a process with a relatively large time delay:

I1 p

bτ =0.86 A t

1-b 21)

Case (2). For τ/θ>8 we see from Figure 8 that the ratio θ/tp varies between 0.25 (for

τ/θ=100 with overshoot=0.1) and 0.36 (for τ/θ=8 with overshoot 0.6). We select to use

the average value θ= 0.305tp which is only 15% lower than 0.36 (the worst case). Also

note that for the intermediate overshoot of 0.3, the ratio θ/tp varies between 0.30 and

0.32. In summary, we have for a lag-dominant process

I2 pτ = 2.44 t (22)

Conclusion. For the nominal choice τc=θ, the integral time τI is obtained as the

minimum of the above two values:

I I1 I2 p p

bτ = min( , ) min 0.86A t , 2.44 t

1-b

(23)

Remark. In effect, we are here estimating the effective delay θ from tp only, by using

θ=0.43tp and θ= 0.305tp for obtaining τI1 and τI2, respectively. It is possible get a better

fit to the SIMC-value of τI by making θ also a function of, for example, the overshoot

and the b-value, but we have here chosen to keep it simple.

5. Final choice of the controller settings (detuning)

So far we have derived nominal controller settings that correspond to a closed-loop time

constant equal to the effective delay (τc=θ). However, in many cases one may want to

use less aggressive (detuned) settings (τc>θ), or one may even want to speed up the

response (τc<θ). To this end, we want to introduce a detuning factor F, where F>1

corresponds less aggressive settings and F<1 to more aggressive settings [12].

To find out how the factor F should be included in the expressions for the controller

gain and integral time we go back to the SIMC settings in Eqs. (5) and (6). The nominal

SIMC setting for τc=θ considered so far (denoted * for clarity) are

Kc* = (0.5 τ / kθ), τI

* = min(τ, τI2*) where τI2

* = 8 θ

The general formulas (5) and (6) can then be rewritten as

Kc = Kc* / F (24)

τI = min(τ, τI2* F) (25)

where

F = (τc + θ)/ 2θ (26)

Note that τc=θ gives F=1. Formulas (24) and (25) can now be used to generalize the

proposed settings in Eqs.(14) and (23), which are based τc=θ. In conclusion, the final

tuning formulas for the proposed “Setpoint Overshoot Method” method are:

c c0K = K A F (27)

I p p

bτ =min 0.86A t , 2.44 t F

1-b

(28)

where 2A= 1.152(overshoot) - 1.607(overshoot) + 1.0 and F is a detuning parameter.

F=1 gives the “fast and robust” SIMC settings corresponding to τc=θ, see Eq.(26). To

detune the response and get more robustness one selects F>1, but in special cases one

may select F<1 to speed up the closed-loop response.

6. Analysis and Simulation

Closed-loop simulations have been conducted for 33 different processes and the

proposed tuning procedure provides in all cases acceptable controller settings with

respect to both performance and robustness.

For each process, PI-settings were obtained based on step response experiments with

three different overshoot (about 0.1, 0.3 and 0.6) and compared with the SIMC settings.

The closed-loop performance is evaluated by introducing a unit step change in both the

set-point and load disturbance i.e, (ys=1 and d=1).

Output performance (y) is quantified by computing the integrated absolute error,

0

IAE= dtsy y

. Manipulated variable usage is quantified by calculating the total

variation (TV) of the input (u), which is the sum of all its moves up and down. If we

discretize the input signal as a sequence [u1,u2,u3….,ui…] then i+1 i

i=1

TV= u -u

. Note also

that TV is the integral of the absolute value of the derivative of the input, 0

duTV= dt

dt

, so

TV is a good measure of the smoothness. To evaluate the robustness, we compute the

maximum closed-loop sensitivity, defined as s ωM =max 1/[1+g c(jω)] . Since Ms is the

inverse of the shortest distance from the Nyquist curve of the loop transfer function to

the critical point (-1, 0), a small Ms-value indicates that the control system has a large

stability margin. We want IAE, TV and Ms all to be small, but for a well tuned

controller there is a trade-off, which means that a reduction in IAE implies an increase

in TV and Ms (and vice versa).

The results for the 33 example processes, which include the 15 examples (E1-E15) from

Skogestad [6] and some additional examples with oscillating and unstable plant

dynamics, are listed in Table 1. For first-order processes (E14, E15, E16), a small delay

must be added (E14a, E15a, E16a) to be able to get the closed-loop overshoot needed to

apply the proposed method. All results are without detuning (F=1). The complete

simulation results for all cases are available in a technical report [1].

As expected, when the method is tested on first-order plus delay processes, similar to

those used to develop the method, the responses are similar to the SIMC-responses,

independent of the value of the overshoot. Typical cases are E17 (first-order with

delay), E21 (pure time delay) and E24 (integrating with delay); see Figures 9-11.

For models that are not first-order plus delay (typical cases are E1, E5 and E8; see

Figures 12-14), the agreement with the SIMC-method is best for the intermediate

overshoot (around 0.3). A small overshoot (around 0.1) typically give "slower" and

more robust PI-settings, whereas a large overshoot (around 0.6) gives more aggressive

PI-settings. In some sense this is good, because it means that a more "careful" step

response results in more "careful" tunings. Also note that the user always has the option

to use the detuning factor F to correct the final tunings. Note that the proposed method

works well on some unstable (E33; Figure 15) and oscillating processes (E30, E31,

E32) where there are no PI-rule for the SIMC method.

The effect of using the detuning factor F is illustrated in Figure 16 using a simple

second order process (case E1). As expected, using F>1 results in more robust controller

settings.

7. Derivative action (PID control)

The tuning rules derived in this paper are for PI control (3). In theory, one may better

robustness and/or better output performance by adding derivative action (PID control).

There are many PID implementations, and we here consider the “classical” PID

controller on cascade form,

PID c / )I

D

D

1 + τ s 1 (

1c s =K 1+

τ s N s

(29)

Here, τD is the derivative time and the filter parameter N is typically around 10

(O’Dwyer [3]). Because the addition of derivative action comes at the expense of a

more complex controller, more sensitivity to measurement noise and more input usage,

Skogestad [6] recommends that derivative action (PID control) is only justified for

dominant second-order process,

- s

1 2

eg(s)=

(τ s+1) (τ s+1)

(30)

where “dominant” means that the second-order time constant (τ2) is larger than the

effective time delay θ. Of the 33 processes in Table 1, this would apply to cases E1, E2,

E3, E5, E6, E8, E10 and E27. One particular example is the double integrating process

(E27, τ2=∞), which actually can be stabilized only if we add derivative action to the PI

controller.

Is it possible to use derivative action with the closed-loop tuning approach in this paper?

Yes, it is, but one cannot just “add on” derivative action to the PI-controller, because

also the controller gain and integral time needs to be changed to achieve the benefit of

adding derivative action.

We suggest adding the derivative action beforehand by using a PD controller during

setpoint experiment,

D

DPD c01 τ s

1 ( / )c s =K N s

(31)

The idea is to include the derivative term during the experiment and then include the

same term in the final controller. It is like designing a PI controller for a modified

process. Two disadvantages with this approach are:

1. We must make sure that the setpoint is differentiated. The “problem” is

that in order to avoid the “derivative kick”, many industrial PID

implementations do not differentiate the setpoint (see Figure 17), and this

will give a too small overshoot in the PD setpoint experiment. There are

two fixes to this problem: (a) Temporarily use a PD-implementation with

setpoint differentiation. (b) Alternatively, and this is better because it

gives less input usage, record the “differentiated” output signal

D/ )D

1 τ s1 (( ) sD N sy s y

(see Figure 17) during the experiment.

2. We must decide on a value for the derivative time constant (τD) before

the setpoint experiment.

What value should one select for τD?

With the cascade PID form in (29), Skogestad [6] recommends selecting τD equal to the

dominant second-order time constant, i.e., τD = τ2. However, this requires that we

already have some process knowledge.

In the absence of process knowledge, a reasonable lower starting value is

τ2 = 0.27 tp (32)

where tp is the peak time from an initial P-only setpoint experiment. The starting value

is derived as follows: If the true model was second-order with a delay θ0, then from the

half rule (Skogestad [6]) the effective delay for a first-order with delay model is θ = θ0 +

τ2/2, which gives τ2 = 2(θ - θ0). We do not know the original time delay θ0, but it should

be smaller than τ2 to have a benefit of derivative action. To get a lower value for τ2 (and

be conservative), let us assume θ0 = τ2, which gives τ2 = 0.67 θ. This value works well

also when the process is not dominant second order, because it is only slightly higher

than the derivative time 0.5 θ recommended to counteract the time delay in a first-order

process (e.g., Rivera et al. [5]). Next, from Figure 8 the effective time delay θ is

between 0.3tp and 0.5tp, so let us use θ = 0.4tp, and we finally obtain τ2 = 0.67 θ = 0.27 tp.

When performing the setpoint experiment with the PD-controller, one should monitor

the value of tp (with approximately the same overshoot) and make sure that this it is

significantly reduced compared to the P-only setpoint experiment. If this is not the case

then the expected benefits of adding derivative action are small.

Proposed approach for PID control. The procedure for the setpoint experiment is

unchanged, expect that we use a PD-controller.

Step 1(D). Use a PD-controller (31) and select a value for the derivative time τD. A

good value is τD = τ2 (second time constant), but if this is not known then a lower

starting value is τD = 0.27 tp, where tp is the time to reach the peak using a P-only

controller. Make sure that (a) the PD controller is set up such that the setpoint is

differentiated or (b) record the differentiated output yD.

Step 2. Adjust Kc0 to get a setpoint overshoot between 10% and 60% (this step is

unchanged, except that we use a PD-controller).

Step 3. Collect data for the setpoint experiment (unchanged).

The recommended formulas for Kc and τI are also unchanged, but note that this assumes

that we use the “cascade” PID controller in (29). To use the common “ideal” PID

controller,

Dc

I D

τ' s1c s =K' 1+

τ' s 1 (τ' /N)s

(33)

we must modify the cascade settings by a factor c = 1+ τD/τI by using the following

translation formulas (e.g. Skogestad [6]) (the effect of the filter parameter N has here

been neglected, i.e., N=∞ is assumed):

K’c = c Kc , τ’I = c τI , τ’D = τD / c (34)

Example PID control. We consider a third-order integrating process, 2

1( )

( 1)g s

s s

(case E8). From the “half rule” of Skogestad [6] this can be approximated by a second-

order integrating process s

2

eg s

s (τ s+1)

with θ = 0.5 and 2τ = 1.5. Since τ2 is 3

times larger than the effective time delay θ, this is a “dominant” second-order process,

so we expect significant benefits from adding derivative action.

We now follow the procedure for the setpoint experiment.

Step 1(D). Switch the controller to PD mode. With P-only control we found tp = 6.19

for an overshoot of 30.7% (Table 1 and Table 2) and based on this, we select τD = 0.27 ·

6.19 = 1.67, which is close to the value τD = 1.5 recommended by Skogestad [6] for this

process. The filter parameter for the PD-controller (31) is set at N= 10.

Step 2. Setpoint experiments with overshoots of about 30% are shown in Figure 18 for

the P-controller (Kc0=0.58, τD=0) and PD-controller (Kc0=1.54, τD = 1.67). The time to

reach the first peak (tp) is reduced from 6.2 (P-control) to 2.3 (PD-control) which

indicates a large benefit of adding derivative action.

Step 3. The results of the setpoint experiment are summarized in Table 2. The resulting

cascade PID settings with F=1 are Kc=0.945, τI=5.49 min and τD = 1.67 min. If one uses

the ideal form (33), then one must translate the values using (34) with c=1.30.

In the closed-loop simulations in Figure 19, we use the PID structure in Figure 17 with

N=10 and no differentiation of the setpoint. Comparing the closed-loop responses for PI

and PID-control in Figure 19, we note that there is, as expected, a large benefit of

adding derivative action for this process. This holds both for the setpoint and

disturbance responses. The system is also more robust (sensitivity peak Ms is reduced

from 1.75 to 1.49), but the input usage is somewhat larger and the sensitivity to

measurement noise (not shown in the simulations) is larger

8. Industrial verification

The proposed tuning method for PI control has been verified industrially at the Statoil

refinery at Mongstad, Norway. They report [16] that the algorithm works well in most

cases and they see it as a great advantage to be able to do the tuning in closed loop. The

detuning factor was set to F=1 in most cases. They have previously used the SIMC

method [6] based on an open-loop step test, but this took longer time than the closed-

loop test.

Results for the pressure control loop for the crude prefractionator are shown in Figure

20 (setpoint experiment) and Figure 21 (resulting closed-loop PI response). The output

y (CV) is the pressure [barg] and the input u (MV) is the valve position [%]. The

responses are a bit jerky because of infrequent updates of the pressure measurements.

From the setpoint experiment with Kc0 = 35 we obtain the following data (Figure 20):

tp = 541 s – 516 s = 25 s = 0.417 min

y0 = 1.805 barg

ys = 1.700 barg

yp = 1.671 barg

yu = 1.741 barg

and we obtain the output changes

Δys = |ys-y0| = 0.105 barg

Δyp = |yp-y0| = 0.134 barg

Δyu = |yu – y0| = 0.064 barg

To save time, the experiment was not run to steady-state, but from (10) the predicted

steady-state change is

Δy∞ = 0.45 (Δyp+Δyu) = 0.45 · (0.134 + 0.064) = 0.089 barg

From this we get

Overshoot = (Δyp-Δy∞)/Δy∞ = 0.506

Steady-state ratio, b = Δy∞/Δys = 0.847

With this overshoot, we find from (15) the gain ratio A=0.48, and with a selected

detuning factor F = 1.2, the final settings are from (27) and (28):

Kc = Kc0*A/F = 14.0

τI = min (0.95, 1.22) = 0.95 min

The final closed-loop response with PI control is shown in Figure 21. The response is

good, except for some apparent jerky behavior caused by the infrequent the

measurement update.

9. Discussion: Comparison with related approaches

An obvious alternative to the proposed method is a two-step procedure where one first

identifies an open-loop model g (with parameters k, θ and τ) from the closed-loop

setpoint experiment, and then obtains the PI or PID controller using standard tuning

rules (e.g., the SIMC rules [6]). In fact, from (18), we can obtain the following estimate

of the process gain k,

1

c0

bk = K

(1-b) (35)

Similarly, from (21), we have the following estimate of the dominant time constant τ,

p

bτ = 0.86 A(overshoot) t

1-b (36)

where A(overshoot) is given in (16). Finally, for the effective time delay θ, we have the

following estimate from case (2):

θ= 0.305tp (37)

One may use (35)-(37) also with other tuning rules, but note these estimates, and in

particular A(overshoot) from (16), were derived with the goal of matching the SIMC PI

settings. Nevertheless, the estimates (35)-(37) may be useful, for example, for gaining

insight or when comparing or combining with information obtained from an open-loop

experiment. However, if the goal is to use the setpoint experiment to obtain PI settings,

then we recommend the one-step procedure with PI-settings obtained from (27) and

(28), rather than the two-step procedure where one first obtains the open-loop model g.

A two-step procedure, based on a closed-loop setpoint experiment with a P-controller,

was originally proposed by Yuwana and Seborg [10]. They identified a first-order with

delay model by matching the closed-loop setpoint response with a standard oscillating

second-order step response that results when the time delay is approximated by a first-

order Pade approximation. They identified from the setpoint response the first

overshoot, first undershoot and second overshoot, but the method may be modified to

not use the second overshoot, as in the present paper. Yuwana and Seborg [10] then

used the Ziegler-Nichols [7] tuning rules, which as mentioned in the introduction may

give rather aggressive setting. We tried using the SIMC rules instead. This improves the

performance, but nevertheless we found (see [11] for details) that the two-step approach

of Yuwana and Seborg [10] gives results slightly inferior to the one-step approach

proposed in this paper.

Veronesi and Visioli [13] recently published another two-step approach, where the idea

is to assess and possibly retune an existing PI controller. From a closed-loop setpoint or

disturbance response using the existing PI controller, they identify a first-order with

delay model and time constant and use this to assess the closed-loop performance. If the

performance is worse that should be expected, the controller is retuned, for example,

using the SIMC method. So far the method has only been developed for integrating

processes.

In another recent paper, Seki and Shigemasa [14] propose to retune the controller based

comparing closed-loop responses obtained with two different controller settings.

9. Conclusion

A simple and new approach for PI controller tuning has been developed. It is based on a

single closed-loop setpoint step experiment using a P-controller with gain Kc0. The PI-

controller settings are then obtained directly from following three data from the setpoint

experiment:

Overshoot = (Δyp - Δy∞) /Δy∞

Time to reach overshoot (first peak), tp

Relative steady state output change, b = Δy∞/Δys.

If one does not want to wait for the system to reach steady state, one can use the

estimate Δy∞ = 0.45(Δyp + Δyu) where Δyu is the output change at the first undershoot.

The proposed PI tuning formulas for the proposed “Setpoint Overshoot Method”

method are:

c c0K = K A F

I p p

bτ =min 0.86A t , 2.44t F

1-b

where 2A= 1.152(overshoot) - 1.607(overshoot) + 1.0 .

The factor F is a detuning parameter. A good trade-off between robustness and speed of

response is achieved with F=1, but one may use F>1 to get a smoother response with

more robustness and less input usage.

The Setpoint Overshoot Method works well for a wide variety of the processes typical

for process control, including the standard first-order plus delay processes as well as

integrating, high-order, inverse response, unstable and oscillating process. The method

gives a PI controller, but for dominant second-order processes where derivative action

may give large benefits, one can use a PD-controller in the setpoint experiment, to end

up with a PID controller.

Compared to the classical Ziegler-Nichols closed-loop method [7], including its relay

tuning variants [3], the proposed overshoot method is faster and simpler to use and also

gives better settings in most cases. The new overshoot method is therefore well suited

for use in the process industries as has already been verified.

Appendix A

The proposed method requires one to obtain the steady-state output change (Δy∞) for the

setpoint response, but it may take some time for the response to settle to the new steady

state. To avoid this, we propose to estimate Δy∞ from only Δyp (first peak) and Δyu

(undershoot). Yuwana and Seborg (1982) derived an estimate of Δy∞ that makes use of

also the second peak, but achieve sufficient accuracy without this information. Since

Δy∞ must lie between Δyp and Δyu, a first try is to consider the average. In Figure 22 we

plot Δy∞ as a function of the average [(Δyp+Δyu)/2] for 15 first-order with delay process

using 5 different overshoot (0.2, 0.3, 0.4, 0.5, 0.6). Somewhat surprisingly, we find for

the 75 cases that there is almost a linear relationship with a coefficient of 0.895

corresponding to

Δy∞=0.895(Δyp+Δyu)/2≈ 0.45(Δyp+Δyu)

as given in Eq. (10). In Table 3, the correlation has been further tested on the 33

processes from Table 1. For an overshoot of about 30% we find that the deviation is less

than 1 % in most cases; the main exceptions are for processes with overshoot in the

open-loop response (positive zeros; E12, E13, E24) and oscillations (E30, E31) where it

underpredicts the b-value by as much as to -18% (E23).

Acknowledgement. Arne Skage at the Statoil Mongstad refinery provided the industrial

data. Comments from Ivar J. Halvorsen and Vinay Kariwala are gratefully

acknowledged.

References

[1] L. D. Desborough, R. M. Miller, Increasing customer value of industrial control

performance monitoring—Honeywell’s experience. Chemical Process Control –VI

(Tuscon, Arizona, Jan. 2001), AIChE Symposium Series No. 326. Volume 98,

USA, 2002.

[2] M. Kano, M. Ogawa, The state of art in advanced process control in Japan, IFAC

symposium ADCHEM 2009, Istanbul, Turkey, July 2009.

[3] A. O’Dwyer, Handbook of PI and PID Controller Tuning Rules, Imperial College

Press, (2006).

[4] D. E. Seborg, T. F. Edgar, D. A. Mellichamp, Process Dynamics and Control, 2nd

ed., John Wiley & Sons, New York, U.S.A. (2004).

[5] D. E. Rivera, M. Morari, S. Skogestad, Internal model control. 4. PID controller

design, Ind. Eng. Chem. Res., 25 (1) (1986) 252–265.

[6] S. Skogestad, Simple analytic rules for model reduction and PID controller tuning,

Journal of Process Control 13 (2003), 291–309.

[7] J. G. Ziegler, N. B. Nichols, Optimum settings for automatic controllers. Trans.

ASME, 64 (1942) 759-768.

[8] K. J. Åström, T. Hägglund, Automatic tuning of simple regulators with

specifications on phase and amplitude margins, Automatica 20 (1984) 645–651.

[9] B.D. Tyreus, W.L. Luyben, Tuning PI controllers for integrator/dead time

processes, Ind. Eng. Chem. Res. (1992) 2625–2628.

[10] M. Yuwana, D. E. Seborg, A new method for on-line controller tuning, AIChE

Journal 28 (3) (1982) 434-440.

[11] M. Shamsuzzoha, S. Skogestad. Internal report with additional data for the setpoint

overshoot method (2010). Available at the home page of S. Skogestad.

http://www.nt.ntnu.no/users/skoge/. MAKE AVAILABLE AT JPC?!

[12] W.L. Luyben, Simple method for tuning SISO controllers in multivariable systems,

Ind. Eng. Chem. Process Des. Dev. 25 (3) (1986) 654-660.

[13] M. Veronesi, A.Visioli, Performance assessment and retuning of PID controllers

for integral processes, Journal of Process Control, 20 (2010), 261-269.

[14] H. Seki, T. Shigemasa, Retuning oscillatory PID control loops based on plant

operation data, Journal of Process Control, 20 (2010), 217-227.

[15] T. S. Schei, A method for closed loop automatic tuning of PID controllers,

Automatica, 20 (3) (1992), 587-591.

[16] A. Skage, Personal email communication (2010).

Table 1: PI controller setting for proposed method (F=1) and comparison with SIMC method (τc=θeffective obtained using half rule)

P-control setpoint experiment Resulting PI-controller

Setpoint Load disturbance

Case Process model

Kc0 overshoot tp b Kc τI Ms IAE (y) TV(u) overshoot(y) IAE (y) TV(u) peak value (y)

5.0 0.127 0.710 0.833 4.074 1.732 1.33 0.44 8.0 0.03 0.43 1.17 0.20 15.0 0.322 0.393 0.937 9.031 0.958 1.74 0.30 23.72 0.29 0.11 1.81 0.11

40.0 0.508 0.230 0.976 19.23 0.561 2.62 0.33 74.74 0.53 0.03 3.05 0.06

E1

1

1 0.2 1s s

SIMC - - - 5.50 0.80 1.56 0.36 12.65 0.23 0.15 1.55 0.15 0.85 0.131 5.31 0.46 0.688 3.141 1.41 4.56 1.20 0 4.57 1.01 0.60 1.50 0.303 4.450 0.60 0.929 3.562 1.56 3.83 1.76 0 3.83 1.09 0.56 2.50 0.567 3.886 0.714 1.148 3.836 1.73 3.43 2.43 0.04 3.35 1.28 0.54

E2 3

0.3 1 0.08 1

2 1 1 0.4 1 0.2 1 0.05 1

s s

s s s s s

SIMC - - - 0 .85 2.50 1.66 3.56 1. 90 0.1 2.97 1.26 0.57 2.75 0.107 0.711 0.846 2.315 1.734 1.48 0.55 4.74 0.03 0.75 1.20 0.35 5.0 0.314 0.527 0.909 3.043 1.287 1.70 0.43 7.16 0.17 0.43 1.48 0.30 9.50 0.599 0.402 0.950 4.283 0.981 2.18 0.45 12.88 0.36 0.23 2.14 0.24

E3 2

2 15 1

20 1 1 0.1 1

s

s s s

SIMC - - - 2.33 1.05 1.55 0.47 4.96 0.12 0.45 1.29 0.34 0.50 0.10 6.60 0.333 0.426 2.416 1.36 5.66 1.0 0 5.66 1.0 0.69 1.25 0.304 5.250 0.556 0.773 3.489 1.56 4.50 1.49 0 4.50 1.09 0.62 2.50 0.598 4.414 0.714 1.128 4.281 1.85 3.99 2.56 0.06 3.80 1.44 0.56

E4

41

1s

SIMC - - - 0.30 1.50 1.46 5.59 1.15 0.05 5.40 1.10 0.71 3.0 0.104 0.922 0.750 2.534 2.005 1.33 0.79 4.76 0 0.79 1.08 0.28 6.50 0.292 0.615 0.867 4.093 1.50 1.59 0.46 9.13 0.13 0.37 1.42 0.21 15.0 0.599 0.429 0.938 6.766 1.048 2.18 0.47 20.96 0.37 0.16 2.28 0.16

E5

1

1 0.2 1 0.04 1 0.008 1s s s s

SIMC - - - 3.72 1.10 1.59 0.45 8.26 0.16 0.30 1.45 0.22 0.40 0.110 7.636 1.0 0.335 18.63 1.48 6.14 0.75 0.23 55.66 1.46 2.91 0.80 0.301 4.987 1.0 0.496 12.169 1.77 4.74 1.29 0.37 24.51 1.81 2.13 2.0 0.576 3.199 1.0 0.913 7.805 2.73 4.71 3.57 0.58 8.55 3.08 1.32

E6

2

2

0.17 1

1 0.028 1

s

s s s

SIMC - - - 0.296 13.5 1.48 6.50 0.67 0.28 45.61 1.55 3.09

25

P-control setpoint experiment Resulting PI-controller Setpoint Load disturbance

Case Process model Kc0 overshoot tp b Kc τI Ms

IAE(y) TV(u) overshoot(y) IAE(y) TV(u) peak value(y)

0.22 0.111 6.717 0.180 0.184 1.062 1.96 7.42 1.41 0.16 8.92 1.63 1.06 0.40 0.309 5.979 0.286 0.245 1.263 2.13 7.04 1.57 0.21 8.62 1.83 1.08

0.60 0.604 5.595 0.375 0.270 1.298 2.28 7.13 1.73 0.26 8.75 2.01 1.10

E7

32 1

1

s

s

SIMC - - - 0.214 1.50 1.66 6.78 1.05 0.02 8.34 1.28 1.03 0.32 0.106 8.985 1.0 0.270 21.923 1.51 7.22 0.60 0.23 81.26 1.46 3.62 0.58 0.307 6.188 1.0 0.357 15.10 1.75 6.21 0.90 0.35 42.33 1.72 2.92 1.15 0.610 4.492 1.0 0.516 10.961 2.30 5.61 1.67 0.52 21.26 2.50 2.25

E8

21

1s s

SIMC - - - 0.330 12.0 1.76 6.45 0.84 0.38 36.36 1.78 3.02 0.50 0.116 4.55 0.333 0.415 1.622 1.43 3.91 1.0 0 3.91 1.0 0.73 1.0 0.321 3.85 0.50 0.603 1.995 1.58 3.31 1.27 0 3.31 1.04 0.69 1.70 0.623 3.453 0.63 0.758 2.251 1.74 3.03 1.69 0.03 2.97 1.19 0.67

E9

21

se

s

SIMC - - - 0.50 1.50 1.61 3.38 1.32 0.06 3.14 1.15 0.71 4.50 0.11 11.65 0.818 3.769 28.426 1.42 7.54 7.32 0.01 7.54 1.10 0.21 8.0 0.301 8.425 0.889 4.966 20.557 1.62 5.92 10.99 0.14 4.14 1.34 0.18 14.0 0.594 6.594 0.933 6.326 16.088 1.90 6.28 16.42 0.28 2.54 1.72 0.16

E10

20 1 2 1

se

s s

SIMC - - - 5.25 16.0 1.72 6.34 12.31 0.22 3.05 1.49 0.17 0.70 0.119 16.62 0.412 0.577 8.25 1.41 14.3 1.08 0 14.32 1.01 0.65 1.40 0.344 13.67 0.583 0.817 9.602 1.59 11.72 1.60 0 11.78 1.09 0.61 2.20 0.608 12.28 0.687 0.987 10.423 1.74 10.78 2.11 0.03 10.59 1.25 0.58

E11 2

1

6 1 2 1

ss e

s s

SIMC - - - 0.70 7.0 1.63 11.5 1.59 0.08 10.14 1.20 0.62 12.0 0.128 0.89 0.923 9.766 2.171 1.81 0.90 23.40 0.08 0.23 1.32 0.09 15.0 0.308 0.836 0.938 9.220 2.040 1.75 0.92 21.54 0.06 0.23 1.26 0.09 20.0 0.609 0.792 0.952 8.971 1.933 1.72 0.93 20.79 0.06 0.22 1.24 0.09

E12

0.36 1 3 1

10 1 8 1 1

ss s e

s s s

SIMC - - - 7.40 1.0 1.66 1.07 18.28 0.19 0.15 1.39 0.10

26



Case Process model

Kc0 overshoot tp b Kc τI Ms

IAE(y) TV(u) overshoot(y) IAE(y) TV(u) peak value(y) 4.0 0.142 2.25 0.80 3.179 5.490 1.87 2.70 7.44 0.04 1.74 1.28 0.23 4.75 0.302 2.20 0.826 2.943 5.367 1.76 2.88 6.60 0.04 1.85 1.20 0.23

6.0 0.570 2.143 0.857 2.75 5.068 1.68 3.03 6.01 0.05 1.88 1.15 0.23

E13

2 1

10 1 0.5 1

ss e

s s

SIMC - - - 2.88 4.50 1.74 2.86 6.56 0.06 1.62 1.20 0.23 - - - - - - - - - - - - - - - - - -

No oscillation with P-controller, proposed method does not apply

- - - - - - - - -

E14 1s

s

SIMC - - - 0.50 8.0 2.0 2.92 2.04 0.20 17.3 3.40 1.87 0.65 0.106 1.887 1.0 0.548 4.604 2.48 2.74 2.31 0.40 9.91 3.81 2.01 0.70 0.285 1.655 1.0 0.445 4.037 2.01 3.58 1.74 0.45 11.63 3.40 2.17 0.75 0.558 1.584 1.0 0.347 3.866 1.84 4.89 1.28 0.47 16.56 3.00 2.42

E14 (a)

0.11 ss e

s

SIMC - - - 0.455 8.80 1.95 3.38 1.58 0.21 20.69 2.69 2.09 - - - - - - - - - - - - - - - - - -


- - - - - - - - -

E15

1

1

s

s

SIMC - - - 0.50 1.0 2.0 2.0 1.02 0 2.85 3.0 0.89 0.42 0.107 1.75 0.296 0.353 0.532 2.88 2.83 2.56 0.44 3.96 3.25 1.20 0.51 0.31 1.55 0.338 0.314 0.418 3.90 4.12 3.88 0.67 5.26 4.41 1.26 0.60 0.605 1.50 0.375 0.270 0.348 4.64 5.24 4.83 0.77 6.36 5.29 1.27

E15 (a)

0.21

1

ss e

s

SIMC - - - 0.417 1.0 1.88 1.86 1.02 0 3.26 2.25 1.03 - - - - - - - - - - - - - - - - - -


- - - - - - - - -

E16

1

1s

SIMC - - - τc=θeffective=0 so SIMC-rule does not apply (gives infinite controller gain)

27



Case Process model

Kc0 overshoot tp b Kc τI Ms

IAE(y) TV(u) overshoot(y) IAE(y) TV(u) peak value(y) 12.0 0.123 0.217 0.923 9.833 0.530 1.61 0.15 21.49 0.12 0.054 1.24 0.092 16.0 0.309 0.174 0.941 9.819 0.425 1.63 0.16 22.08 0.17 0.043 1.32 0.091

22.50 0.612 0.146 0.957 10.082 0.355 1.68 0.17 23.55 0.22 0.035 1.42 0.090

E16 (a)

0.05

1

se

s

SIMC - - - 10.0 0.40 1.65 0.16 22.83 0.18 0.04 1.36 0.091 2.75 0.10 3.60 0.733 2.338 7.240 1.50 3.09 4.49 0 3.09 1.0 0.30 4.0 0.298 3.049 0.80 2.494 6.538 1.56 2.62 4.96 0 2.62 1.04 0.29 5.75 0.599 2.705 0.852 2.592 6.030 1.60 2.33 5.29 0 2.33 1.07 0.29

E17

5 1

se

s

SIMC - - - 2.50 5.0 1.59 2.17 5.16 0.04 2.0 1.08 0.29 0.50 0.109 2.75 0.333 0.419 0.991 1.47 2.39 1.06 0.01 2.37 1.01 0.74 0.90 0.326 2.40 0.474 0.538 1.111 1.58 2.09 1.23 0.01 2.06 1.03 0.72 1.35 0.593 2.25 0.575 0.610 1.181 1.66 1.99 1.39 0.02 1.94 1.08 0.72

E18

1

se

s

SIMC - - - 0.50 1.0 1.59 2.17 1.26 0.04 2.06 1.08 0.72 0.12 0.113 2.0 0.107 0.10 0.172 1.76 2.16 1.26 0.11 2.19 1.25 0.99 0.30 0.292 2.0 0.231 0.189 0.325 1.67 1.88 1.12 0.05 1.87 1.10 0.99 0.60 0.590 2.0 0.375 0.272 0.467 1.66 1.72 1.01 0 1.72 1.01 0.99

E19

0.2 1

se

s

SIMC - - - 0.10 0.20 1.59 2.17 1.09 0.04 2.16 1.08 0.99 0.10 0.10 2.0 0.091 0.085 0.146 1.70 2.01 1.17 0.08 2.01 1.17 1.0 0.30 0.30 2.0 0.231 0.187 0.321 1.61 1.74 1.02 0.01 1.74 1.01 1.0 0.60 0.60 2.0 0.375 0.270 0.464 1.64 1.72 1.01 0 1.72 1.00 1.0

E20

20.05 1

se

s

SIMC - - - 0.037 0.075 1.59 2.22 1.09 0.04 2.22 1.08 1.0 0.10 0.10 2.0* 0.091 0.085 0.146 1.60 1.85 1.09 0.04 1.85 1.08 1.0 0.30 0.30 2.0 0.231 0.187 0.321 1.53 1.72 1.07 0 1.72 1.02 1.0 0.60 0.60 2.0 0.375 0.270 0.465 1.59 1.72 1.16 0 1.72 1.14 1.0

E21 se

SIMC - - - Kc=0,Kc/τI=0.50 1.59 2.17 1.08 0.04 2.17 1.08 1.0

28

P-control setpoint experiment Resulting PI-controller Setpoint Load disturbance

Case Process model Kc0

overshoot

tp

b

Kc

τI

Ms

IAE(y) TV(u) overshoot(y) IAE(y) TV(u) peak value(y) 0.60 0.118 3.911 0.984 0.496 9.544 1.66 3.79 1.15 0.22 19.24 1.43 1.95 0 .80 0.301 3.293 0.988 0.496 8.034 1.68 3.79 1.18 0.25 16.19 1.50 1.94

1.10 0.598 2.913 0.991 0.496 7.108 1.71 3.79 1.22 0.28 14.33 1.56 1.93

E22 100

100 1

se

s

SIMC - - - 0.50 8.0 1.69 3.78 1.20 0.26 16.0 1.51 1.93 0.22 0.136 2.633 1.0 0.177 6.425 2.14 3.73 0.51 0.13 39.48 1.69 5.35 0.26 0.303 2.563 1.0 0.161 6.255 1.96 3.85 0.43 0.08 42.74 1.51 5.45 0.33 0.595 2.442 1.0 0.149 5.958 1.84 3.96 0.39 0.09 44.50 1.40 5.55

E23

10 1

2 1

ss e

s s

SIMC - - - 0.10 2.0 1.67 3.96 0.30 0.23 24.92 1.46 5.93 0.59 0.108 3.976 1.0 0.495 9.702 1.67 3.98 1.16 0.24 19.6 1.47 1.99 0.80 0.302 3.282 1.0 0.496 8.008 1.70 3.94 1.21 0.28 16.15 1.55 1.97 1.10 0.60 2.909 1.0 0.496 7.098 1.72 3.92 1.24 0.30 14.31 1.61 1.96

E24 se

s

SIMC - - - 0.50 8.0 1.70 3.92 1.22 0.28 16.0 1.55 1.96 0.41 0.117 7.51 1.0 0.339 18.323 1.49 6.09 0.76 0.24 53.93 1.48 2.89 0.80 0.304 4.989 1.0 0.495 12.173 1.77 4.76 1.29 0.37 24.61 1.81 2.14 1.90 0.566 3.305 1.0 0.873 8.064 2.64 4.67 3.39 0.57 9.09 2.99 1.36

E25

2

2

6

1 36

s

s s s

SIMC - - - 0.417 9.60 1.74 5.18 1.07 0.39 23.04 1.82 2.36

-0.13 0.116 14.937 1.0 -0.108 36.447 1.49 12.07 0.25 0.24 338.2 1.49 -9.10 -0.25 0.296 9.685 1.0 -0.156 23.632 1.77 9.46 0.41 0.37 151.3 1.82 -6.80 -0.50 0.554 6.653 1.0 -0.232 16.232 2.31 8.83 0.78 0.53 70.16 2.53 -4.99

E26

1.6 0.5 1

3 1

s

s s

SIMC - - - -0.156 16.0 1.93 9.71 0.45 0.46 102.5 2.08 -6.53 Not possible to stabilize with PI controller

E27 2

se

s

29


Kc τI Ms Setpoint Load disturbance

Case Process model

Kc0 overshoot tp b IAE(y) TV(u) overshoot(y) IAE(y) TV(u) peak value(y)

0.22 0.111 6.717 0.180 0.184 1.062 1.96 7.41 1.41 0.16 8.92 1.63 1.06 0.40 0.309 5.979 0.286 0.246 1.263 2.14 7.04 1.56 0.21 8.62 1.83 1.08

0.60 0.604 5.595 0.375 0.270 1.298 2.28 7.13 1.73 0.26 8.75 2.01 1.10

E28 3

2 1

1

s

s

SIMC - - - 0.214 1.50 1.66 6.78 1.05 0.02 8.34 1.28 1.03 0.17 0.111 12.787 0.145 0.142 1.562 1.71 13.52 1.24 0.10 13.47 1.24 0.95 0.40 0.304 11.987 0.286 0.247 2.547 1.70 11.66 1.17 0.07 11.63 1.18 0.94 0.73 0.595 11.487 0.422 0.330 3.256 1.75 10.59 1.17 0.05 10.55 1.17 0.94

E29

2

5

1

1

ss e

s

SIMC - - - 0.115 1.50 1.57 14.14 1.09 0.04 14.19 1.10 0.95 0.55 0.101 1.63 0.355 0.467 0.643 1.42 1.47 1.09 0.01 1.43 1.03 0.68 1.25 0.322 1.40 0.556 0.752 0.905 1.72 1.26 1.57 0.01 1.23 1.21 0.63 2.30 0.581 1.24 0.697 1.048 1.118 2.21 1.19 2.8 0.05 1.08 1.83 0.58

E30

2

9

1 2 9s s s

SIMC - - - - - - - - - - - - 0.30 0.111 1.55 0.231 0.251 0.334 1.58 1.60 1.24 0.08 1.56 1.25 0.81 0.75 0.31 1.40 0.429 0.460 0.554 2.18 1.53 1.53 0.08 1.60 1.77 0.76 1.10 0.457 3.326 0.524 0.557 1.607 2.13 2.86 1.44 0 2.86 1.66 0.77

E31

2

9

1 9s s s

SIMC - - - No SIMC PI rule

- - - - - - - -

0.07 0.112 18.132 0.387 0.058 8.198 1.46 15.4 0.12 0 143.2 1.08 6.16 0.12 0.301 15.043 0.519 0.074 8.667 1.61 12.74 0.16 0.01 119.4 1.17 5.98 0.18 0.583 13.71 0.618 0.082 8.684 1.70 12.18 0.18 0.05 108.9 1.27 5.91

E32

2 2

22

2 9 2 1 1 e

0.5 1 5 1

ss s s s

s s s


- - - - - - - -

3.10 0.10 4.647 1.476 2.636 10.54 2.12 7.69 9.48 0.87 4.0 2.73 0.56 4.0 0.30 3.671 1.333 2.487 7.852 2.33 7.96 10.15 0.96 3.81 3.12 0.58 5.30 0.607 3.164 1.233 2.379 6.475 2.67 9.03 11.12 1.03 4.18 3.63 0.59

33

5 1

se

s


- - - - - - -

30

* For pure time delay process (E21), obtain tp as end time of peak (or add a small time constant in simulation).

Table 2: P/PD setpoint experiment and resulting PI/PID controller for process 2

1( )

( 1)g s

s s

(E8).

P/PD setpoint experiment Resulting PI/PID controller

Setpoint Load disturbance Case

Kc0 τD overshoot tp b Kc τI Ms

IAE(y) TV(u) overshoot(y) IAE(y) TV(u) peak value(y)

0.58 0 0.308 6.2 1.0 0.357 15.10 1.75 6.21 0.90 0.35 42.33 1.72 2.92 E8 1.54 1.67 (N=10) 0.309* 2.25 1.0 0.945 5.49 1.49 2.68 1.51 0.081** 5.81 1.79 0.81

* The PD controller is with D-action on the setpoint ** The PID controller is without D-action on the setpoint

31

Table 3: % Error in steady-state output Δy∞ when using Δy∞=0.45(Δyp+Δyu)

% Error in Δy∞ (b value)

Case

overshoot ≈0.30 overshoot ≈0.60 E1 -0.3 1.2 E2 -0.6 1.0 E3 -2.9 -2.5 E4 -1.1 -1.3 E5 -0.8 0.04 E6 -1.0 -1.2 E7 0.9 6.7 E8 -0.8 -1.0 E9 -0.1 1.2 E10 -0.5 0.9 E11 0.2 1.0 E12 -8.0 -7.3 E13 -13.9 -13.0 E14a 1.1 9.0 E15a 2.1 10.9 E16a 0.4 4.3 E17 -0.4 1.8 E18 0.4 2.8 E19 -0.4 1.8 E20 -0.6 0.8 E21 -0.6 0.8 E22 -0.3 1.7 E23 -18.2 -18.9 E24 -0.3 1.7 E25 -0.9 -1.1 E26 -0.6 1.5 E28 0.9 6.7 E29 0.03 2.9 E30 -6.2 -8.9 E31 -11.4 -11.2 E32 0.3 4.2 E33 -0.4 1.5

32

py

pt

y sy

t0t

uy

sy

y

0y

Fig. 1. Closed-loop step setpoint response with P-only control.

33

Fig. 2. Block diagram of feedback control system.

34

'cK

'IK

0

1

2

3

4

I I 8

Fig. 3. Scaled proportional and integral gain for SIMC tuning rule.

35

0 5 10 15 20 250

0.25

0.5

0.75

1

1.25

1.5

time

OU

TP

UT

y

overshoot=0.10 (Kc0

=5.64)

overshoot=0.20 (Kc0

=6.87)

overshoot=0.30 (Kc0

=8.0)

overshoot=0.40 (Kc0

=9.1)

overshoot=0.50 (Kc0

=10.17)

overshoot=0.60 (Kc0

=11.26)

setpoint

Fig. 4. Step setpoint responses with various overshoots for first-order plus time delay

process,10 1

seg

s

36

0 5 10 15 20 25 300

0.25

0.5

0.75

1

1.25

time

OU

TP

UT

y

Setpoint

/=1 (Kc0

=0.855)

/=0.4 (Kc0

=0.404)

/=100 (Kc0

=79.9)

/=2 (Kc0

=1.636)

/=5 (Kc0

=4.012)

/=10 (Kc0

=8.0)

/=0.2 (Kc0

= 0.309)

/=0 (Kc0

= 0.3)

Fig. 5. Step setpoint responses with overshoot of 0.3 (30%) for eight first-order plus

time delay processes with τ/θ ranging from 0 to 100 -θse

g= , θ=1τs+1

.

37

0 20 40 60 80 100 1200

10

20

30

40

50

kKc

kKc0

0.10 overshootkK

c=0.8649kK

c0

0.20 overshootkK

c=0.7219kK

c0

0.30 overshootkK

c=0.6259kK

c0

0.40 overshootkK

c=0.5546kK

c0

0.50 overshootkK

c=0.4986kK

c0

0.60 overshootkK

c=0.4526kK

c0

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

Fig. 6. Relationship between P-controller gain kKc0 used in setpoint experiment and corresponding SIMC controller gain kKc.

38

0.1 0.2 0.3 0.4 0.5 0.60.4

0.5

0.6

0.7

0.8

0.9

overshoot (fractional)

A

A=1.152(overshoot)2 -1.607(overshoot)+1.0

Fig. 7. Variation of A with overshoot using data (slopes) from Figure 6.

39

0.1 0.3 0.5 0.60

0.1

0.2

0.3

0.4

0.5

Overshoot

/t p

0.43 (I1)

0.305 (I2)

/=0.1

/=8

/=100

/=1

Fig. 8. Ratio of process time delay (θ) and setpoint overshoot time (tp) as a function of

overshoot for four first-order with delay processes (solid lines). Dotted lines: Values of

θ/tp used in final correlations.

40

0 20 40 60 800

0.25

0.5

0.75

1

1.25

time

OU

TP

UT

y

Proposed method with F=1 (overshoot=0.10)Proposed method with F=1 (overshoot=0.298)Proposed method with F=1 (overshoot=0.599)SIMC (

c==1)

Fig. 9. Responses for PI-control of simple first-order process 5 1

seg

s

(E17). Setpoint

change at t=0; load disturbance of magnitude 1 at t=40.

41

0 6 12 18 24 300

0.5

1

1.5

2

time

OU

TP

UT

y


c==1)

Fig. 10. Responses for PI-control of pure time delay process sg e (E21), Setpoint change at t=0; load disturbance of magnitude 1 at t=15.

42

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

time

OU

TP

UT

y

Proposed method with F=1 (overshoot=0.108)Proposed method with F=1 (overshoot=0.302)Proposed method with F=1 (overshoot=0.60)SIMC (c==1)

Fig. 11. Responses for PI-control of integrating process sg e s (E24), Setpoint change at t=0; load disturbance of magnitude 1 at t=50.

43

0 2 4 6 8 100

0.25

0.5

0.75

1

1.25

1.5

time

OU

TP

UT

y


c=

effective=0.1)

Fig. 12. Responses for PI-control of second-order process

1

1 0.2 1g

s s

(E1),

Setpoint change at t=0; load disturbance of magnitude 1 at t=5.

44

0 5 10 15 200

0.25

0.5

0.75

1

1.25

time

OU

TP

UT

y


c=

effective=0.148)

Fig. 13. Responses for PI-control of high-order process

1

1 0.2 1 0.04 1 0.008 1g

s s s s

(E5), Setpoint change at t=0; load disturbance of

magnitude 1 at t=10.

45

0 40 80 120 160 2000

1

2

3

4

5

time

OU

TP

UT

y


c=

effective=1.5)

Fig. 14. Responses for PI-control of third-order integrating process 2

1

1g

s s

(E8),


46

0 20 40 60 800

0.5

1

1.5

2

time

OU

TP

UT

y

Proposed method with F=1 (overshoot=0.10)Proposed method with F=1 (overshoot=0.30)Proposed method with F=1 (overshoot=0.607)

Fig. 15. Responses for PI-control of first-order unstable process 5 1

seg

s

(E33),


47

0 2 4 6 8 100

0.25

0.5

0.75

1

1.25

time

OU

TP

UT

y

F=1.0 (overshoot=0.322)F=2.0 (overshoot=0.322)F=3.0 (overshoot=0.322)F=0.8 (overshoot=0.322)

Fig. 16. Effect of detuning factor: Responses for PI-control of second-order

1

1 0.2 1g

s s

(E1), Setpoint change at t=0; load disturbance of magnitude 1 at t=5.

F Kc τI Ms 1.0 9.031 0.958 1.74 2.0 4.52 1.92 1.36 3.0 3.01 2.87 1.24 0.8 11.29 0.77 1.96

48

s +

-c

I

1K 1+

τ s

D

D

τ s+1τ

s+1N

D

Fig. 17. Cascade implementation of PID controller without differentiation of setpoint.

49

0 4 8 12 16 200

0.2

0.4

0.6

0.8

1

1.2

1.4

time

OU

TPU

T y

P ControllerPD Controller

Fig. 18. Setpoint experiment with P and PD controller for 3rd order integrating process

(E8). PD controller has filter parameter N=10 and setpoint is differentiated.

50

0 10 20 30 40 50 600

1

2

3

4

time

OU

TPU

T y

0 10 20 30 40 50 60-2

-1

0

1

time

INP

UT

u

PI (Proposed method)PID (Proposed method)

Fig. 19. Response with PI- and PID-controller for for 3rd order integrating process (E8).

PID controller has filter parameter N=10 and setpoint is not differentiated.

51

500 510 520 530 540 550 560 570 580 590 6001.65

1.7

1.75

1.8

1.85

time(second)

barg

500 510 520 530 540 550 560 570 580 590 60030

40

50

60

70

80

time(second)

%

CVCVset

MV

Fig. 20. Industrial verification. Setpoint experiment

52

0 100 200 300 400 500 600 700 800 9001.65

1.7

1.75

1.8

1.85

1.9

time(seond)

barg

0 100 200 300 400 500 600 700 800 90030

40

50

60

70

time(second)

%

MV

CVCVset

Fig. 21. Industrial verification. Final closed-loop PI response

53

Line: Δy∞ = 0.8947(Δyp+ Δyu)/2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

(Δyp+ Δyu)/2

Δy ∞

overshoot=0.6

overshoot=0.5

overshoot=0.4

overshoot=0.3

overshoot=0.2

Fig. 22. Relationship between Δy∞ and (Δyp+Δyu)/2 for 15 first-order with delay process with 5 different overshoots.

the setpoint overshoot method: a simple and fast closed-loop...

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